We analyse the coalescence of invasive cell populations by studying both the temporal and steady behaviour of a system of coupled reaction-diffusion equations. This problem is relevant to recent experimental observations of the dynamics of opposingly directed invasion waves of cells. Two cell types, u and v, are considered with the cell motility governed by linear or nonlinear diffusion. The cells proliferate logistically so that the long-term total cell density, u+v approaches a carrying capacity. The steady-state solutions for u and v are denoted u(s) and v(s). The steady solutions are spatially invariant and satisfy u(s)+v(s)=1. However, this expression is underdetermined so the relative proportion of each cell type u(s) and v(s) cannot be determined a priori. Various properties of this model are studied, such as how the relative proportion of u(s) and v(s) depends on the relative motility and relative proliferation rates. The model is analysed using a combination of numerical simulations and a comparison principle. This investigation unearths some novel outcomes regarding the role of overcrowding and cell death in this type of cell migration assay. These observations have relevance to experimental design and interpretation regarding the identification and parameterisation of mechanisms involved in cell invasion.